How many positive divisors does $24$ have?
We will find the positive divisors of 24 by finding pairs that multiply to 24.  We begin with $1$ and $24$, so our list is $$1 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 24.$$  Checking $2$, we find that $2\cdot 12=24$, so our list becomes $$1 \quad 2 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 12 \quad 24.$$  Checking $3$, we find that $3\cdot 8=24$, so our list becomes $$1 \quad 2 \quad 3 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 8 \quad 12 \quad 24.$$  Checking $4$, we find that $4\cdot 6=24$, so our list becomes $$1 \quad 2 \quad 3 \quad 4 \quad \underline{\hphantom{10}} \quad \dots \quad \underline{\hphantom{10}} \quad 6 \quad 8 \quad 12 \quad 24.$$  Checking $5$, we find that $24$ is not divisible by $5$, and since $6$ is already on our list, we are done.  Thus our final list is $$1 \quad 2 \quad 3 \quad 4 \quad 6 \quad 8 \quad 12 \quad 24.$$  Therefore, we can count the number of numbers in our list to find that $24$ has $\boxed{8}$ positive divisors.